经济系统结构与功能的分形模型——Cobb Douglas生产函数的理论来源及形式推广

General fractal models on structure and function of dynamic economy systems:theoretical foundations and generalized form of the Cobb Douglas' production function

从经济系统的定义及其投入—产出的响应模型出发，导出具有分形性质的弹性公式和生产函数，从而揭示了ＣｏｂｂＤｏｕｇｌａｓ 函数的理论来源并将其推广到一般形式，解决了经济学中一些悬而未决的理论难题－

A reseach is made into the fractals and fractal dimensions of a dynamic economy system dX i/dt=f i(X 1,X 2,…,X n), from which the allometric relationship,X i∝X α ij j,is derived as a dynamic similarity model, where X iandX jare some kind of measurements of the ith and jth elements of the system given,α ij =(dX i/(X idt))/(dX j/(X jdt))=D i/D j,is the scaling factor, in this expression D i is the general dimension of X i and D j dimension of X i. On the other hand, on the condition that the system is fractal,the comprehensive Cobb Douglas's model, y=μ ΠniX σ i i ,can be derived out from the production function, y=k·f(X 1,X 2,…,X n), where k、μ are production coefficients, and σ i=(y·X i)/(X i·y) is the elastic coefficient, which has some nature of fractal dimension and D=|σ i|.Based on the general C D function ,the fractal optimization conditions is derived as follows: σ 1/(P 1X 1)=σ 2/(P 2X 2)=…=σ n/(P nX n), where P i is the generalized price of the ith element (i=1,2,…;j,…,n)